Finite Group Databasebeta

← #10944106

Subgroups — #10944106

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic
2#82786C211yesmaximal, characteristic
3C2×#5050C211yesmaximal, characteristic
4C2×#4795C212yesmaximal, complemented
5C2×#4739C211yesmaximal, characteristic
6C3×#1873C211yesmaximal, characteristic
7#71585C211yesmaximal, characteristic
8#70281C212yesmaximal, complemented
9#69909C211yesmaximal, characteristic
10#68908C212yesmaximal, complemented
11#68905C212yesmaximal, complemented
12#32915—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#944C2211yescharacteristic
17C22×#216C2212yes—
18C2×#926C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×Dic12C2212yes—
21#5516—28no—
22C2×#887C2212yes—
23C2×#887C2212yes—
24C2×#877—22no—
25C2×C4×Dic6—22no—
26C2×C3×#103C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#816C2212yes—
31C2×Dic3⋊C8C2212yes—
32#5050—24no—
33#4982—22no—
34#4977C2211yescharacteristic
35#4973C2211yescharacteristic
36C2×#810C2212yes—
37C2×#810C2212yes—
38#4952—22no—
39#4795—28no—
40#4739—24no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#519—31no—
45C2×#462—32no—
46C2×#451—31no—
47C2×#451—31no—
48#2621—31no—
49#1878—32no—
50#1878—32no—
51#1873—31no—
52#1873D311yescharacteristic
53#1805—32no—
54#1798—31no—
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—21no—
58C22×Dic6—21no—
59C22×Dic6—22no—
60C22×Dic6—24no—
61C22×Dic6—24no—
62C22×Dic6D414yes—
63C22×Dic6C2311yescharacteristic
64C22×Dic6C2312yes—
65C22×C3×C8C2312yes—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×#117—24no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×#217—21no—
76C2×#217D412yes—
77#944—24no—
78C2×#216—24no—
79C2×#216—24no—
80C2×#216—24no—
81#926—24no—
82C2×C4⋊Dic3D412yes—
83C2×C4⋊Dic3C2311yescharacteristic
84#924—22no—
85#924—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95C2×Dic12—24no—
96C2×Dic12—24no—
97C2×Dic12—24no—
98#887—24no—
99#887—24no—
100#887—24no—
101#887D418yes—
102#877—216no—
103C4×Dic6—216no—
104C3×#103—24no—
105C3×C22⋊C8—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109#830—22no—
110#816—24no—
111Dic3⋊C8—24no—
112#810—24no—
113#810—24no—
114C23×Q8—31no—
115C22×Q16—32no—
116C22×Q16—32no—
117C2×C4⋊Q8—31no—
118C2×C22⋊Q8—31no—
119C2×C22⋊Q8C2×D311yescharacteristic
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—62no—
126C2×#117C2×D311yescharacteristic
127#462—68no—
128#451—64no—
129#451—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103C2×D312yes—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#398—62no—
138#397—62no—
139#391—31no—
140#391—31no—
141#382—31no—
142#340—32no—
143#340C2×D312yes—
144#336—32no—
145C24×C3C2×D411yescharacteristic
146C2×C3×Q8—22no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—24no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3C2×D412yes—
166C22×Dic3C2×D412yes—
167C2×Dic6—24no—
168C2×Dic6—24no—
169C2×Dic6—216no—
170C2×Dic6—24no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—216no—
175C2×Dic6—22no—
176C2×Dic6—21no—
177C2×Dic6—22no—
178C2×Dic6—21no—
179C2×Dic6C2×D414yes—
180C2×Dic6C2×D414yes—
181C2×Dic6#4114yes—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D414yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—42no—
192C3×C42—24no—
193#217—24no—
194#217—42no—
195#216—48no—
196C4⋊Dic3—24no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199Dic3⋊C4—28no—
200Dic3⋊C4—28no—
201Dic3⋊C4—42no—
202Dic3⋊C4—44no—
203C4×Dic3—28no—
204C4×Dic3—22no—
205C4×Dic3—28no—
206C4×Dic3C2×D414yes—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210Dic12—48no—
211C22×Q8—34no—
212C22×Q8—32no—
213C22×Q8—31no—
214C22×Q8—64no—
215C22×Q8—64no—
216C22×Q8—62no—
217C22×Q8—61no—
218C22×Q8C22×D311yescharacteristic
219C23×C4—61no—
220C23×C4C22×D311yescharacteristic
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—64no—
224C2×Q16—64no—
225C2×Q16—64no—
226C2×Q16—64no—
227C22×C8—32no—
228C22×C8C22×D312yes—
229C4⋊Q8—64no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—62no—
233C22⋊Q8—62no—
234C4×Q8—38no—
235C4×Q8—616no—
236C4×Q8—64no—
237C4×Q8—64no—
238C4×Q8—64no—
239#117—24no—
240#117—64no—
241#117—616no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—32no—
254C2×C22⋊C4—61no—
255C2×C42—21no—
256C2×C42—31no—
257C2×C42—31no—
258C2×C42—62no—
259C2×C42—62no—
260C4⋊C8—64no—
261#103—24no—
262#103—64no—
263#103—64no—
264#103—64no—
265C22⋊C8—22no—
266C22⋊C8—62no—
267#95—62no—
268#95—62no—
269#95—62no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3#12112yes—
275C23×C3C22×D411yescharacteristic
276C3×Q8—42no—
277C3×Q8—42no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4#12112yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D412yes—
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4C2×#4112yes—
295C2×Dic3—216no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3#12014yes—
310Dic6—216no—
311Dic6—24no—
312Dic6—24no—
313Dic6—24no—
314Dic6—42no—
315Dic6—42no—
316Dic6#12118yes—
317C3×C8—42no—
318C3⋊C8—42no—
319C24D3×D411yescharacteristic
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—34no—
325C2×Q8—34no—
326C2×Q8—34no—
327C2×Q8—616no—
328C2×Q8—64no—
329C2×Q8—64no—
330C2×Q8—616no—
331C2×Q8—64no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—61no—
336C2×Q8—62no—
337C2×Q8—61no—
338C2×Q8—62no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—64no—
347C22×C4—64no—
348C22×C4—64no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4#23512yes—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D412yes—
355C22×C4C23×D311yescharacteristic
356Q16—128no—
357Q16—128no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—42no—
368C4⋊C4—64no—
369C4⋊C4—68no—
370C4⋊C4—68no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—124no—
374C4⋊C4—122no—
375C4⋊C4D3×D414yes—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—64no—
379C22⋊C4—122no—
380C42—24no—
381C42—34no—
382C42—68no—
383C42—68no—
384C42—62no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3C2×#12011yescharacteristic
391C22×C3C2×#12111yescharacteristic
392C22×C3#52511yescharacteristic
393C22×C3#57812yes—
394C22×C3#57812yes—
395C3×C4—22no—
396C3×C4—44no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4C2×#12112yes—
400C3×C4#52512yes—
401Dic3—28no—
402Dic3—24no—
403Dic3—42no—
404Dic3—81no—
405Dic3—81no—
406C23—21no—
407C23—22no—
408C23—22no—
409C23—21no—
410C23#88412yes—
411C23C2×D3×D411yescharacteristic
412Q8—38no—
413Q8—42no—
414Q8—42no—
415Q8—616no—
416Q8—64no—
417Q8—64no—
418Q8—64no—
419Q8—122no—
420Q8—122no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—34no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—616no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—64no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—122no—
447C2×C4—121no—
448C2×C4#88412yes—
449C2×C4C2×#23511yescharacteristic
450C2×C4C2×D3×D412yes—
451C2×C4C2×D3×D411yescharacteristic
452C2×C4D3×#4112yes—
453C8—42no—
454C8—122no—
455C2×C3—42no—
456C2×C3#284711yescharacteristic
457C2×C3#301112yes—
458C2×C3#304411yescharacteristic
459C2×C3#304411yescharacteristic
460C2×C3#312712yes—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—42no—
465C22—42no—
466C22C2×#88411yescharacteristic
467C22D3×#12011yescharacteristic
468C22#565611yescharacteristic
469C22#577112yes—
470C22#577912yes—
471C4—22no—
472C4—44no—
473C4—64no—
474C4—68no—
475C4—81no—
476C4—81no—
477C4—122no—
478C4—241no—
479C4—241no—
480C4#554712yes—
481C4D3×#12112yes—
482C3#3291511yescharacteristic, complemented, Sylow
483C2—42no—
484C2#7493511yescharacteristic
485C2#7886511yescharacteristic
486C2#7897711yescharacteristic
487C2#8023012yes—
488C2#8149912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1873D3
hypocenterC3#32915
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#4795C2semidirect
8. #70281C2semidirect
10. #68908C2semidirect
11. #68905C2semidirect
482. C3#32915semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55162
#827862
C2×#50502
C2×#47952
C2×#47392
C3×#18732
#715852
#702812
#699092
#689082
#689052
#329153